Lecture 29 (Nov. 26-27)
The Divergence Theorem (reading: 16.9)
Theorem: (the “Divergence Theorem”, or “Gauss’s Theorem”) Let E be a 3D solid
region. Let S be the boundary surface of E, with positive (outward) orientation. Let
F be a vector field (with continuous partials in a region containing E). Then
ZZZ
(divF)dV =
E
ZZ
S
F · dS.
See the text for a proof of the divergence theorem for “simple” solid regions.
Example: find the flux of F = h3xy 2 , xez , z 3 i through the pictured surface:
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RR
Example: compute S F · dS, where F = hz 2 x, y 3 /3 + tan z, x2 z + y 2 i, and S is the
unit upper-hemisphere (with “upward” pointing normal).
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Example: what is the flux of
F=
xî + y ĵ + z k̂
(x2 + y 2 + z 2 )3/2
across any closed surface?
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Example: let F = r ⇥ G, where G = hx2 yz, yz 2 , z 3 exy i. Find
as shown:
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RR
S
F · dS, where S is
Example: Let S be a closed surface bounding a solid region E. Let f be a function
with continuous partials. Show
ZZ
ZZZ
(f n)dS =
rf dV
S
E
(note this is a vector equation).
Example: Let S be a surface whose boundary is a curve C, and let f and g be
functions with continuous second partials. Show
Z
ZZ
f rg · dr =
(rf ⇥ rg) · dS.
C
S
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